Average Rate of Change Calculator
Calculate the average rate of change between two points on a function
Use: x^2, sqrt(x), sin(x), cos(x), tan(x), ln(x), e^x, pi
Point 1 (x₁, y₁)
Point 2 (x₂, y₂)
Average Rate of Change
Secant Line Equation
Line through both points
Slope (m)
Rise / Run = Δy / Δx
Calculation Steps
Average Rate of Change
AROC = (f(b) - f(a)) / (b - a)
AROC = Δy / Δx
AROC = (y₂ - y₁) / (x₂ - x₁)
Secant Line
y - y₁ = m(x - x₁)
y = mx + b
m = slope = AROC
Instantaneous Rate
f'(x) = lim[h→0] AROC
Derivative at a point
Applications
• Average velocity
• Average acceleration
• Growth rates
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About Average Rate of Change Calculator
What is Average Rate of Change?
The average rate of change measures how a function's output changes, on average, between two input values. It represents the slope of the secant line connecting two points on a curve.
Formula
Average Rate of Change = (f(b) - f(a)) / (b - a)
Where:
- a = starting x-value
- b = ending x-value
- f(a) = function value at point a
- f(b) = function value at point b
How to Use This Calculator
- Choose Calculation Mode: Select whether to input a function or coordinate points
- Enter Values:
- Function mode: Enter f(x) and the interval [a, b]
- Points mode: Enter coordinates (x₁, y₁) and (x₂, y₂)
- Calculate: Click calculate to get the average rate of change
Understanding Your Results
Interpretation
- Positive value: Function is increasing on average
- Negative value: Function is decreasing on average
- Zero: Function has no net change (may have local changes)
Secant Line
The result is the slope of the secant line. The equation of this line is: y - y₁ = m(x - x₁)
Where m is the average rate of change.
Connection to Derivatives
The average rate of change is related to the derivative (instantaneous rate of change):
- Average rate of change: Slope over an interval
- Derivative: Limit of average rate of change as interval approaches zero
f'(a) = lim[h→0] (f(a+h) - f(a)) / h
Common Applications
| Field | Application |
|---|---|
| Physics | Average velocity |
| Economics | Average cost change |
| Biology | Average population growth rate |
| Business | Average revenue change |
Examples
Example 1: Function
For f(x) = x² on [1, 3]:
- f(1) = 1, f(3) = 9
- Average rate of change = (9 - 1) / (3 - 1) = 4
Example 2: Points
For points (2, 5) and (6, 13):
- Average rate of change = (13 - 5) / (6 - 2) = 2
Frequently Asked Questions (FAQ)
What's the difference between average and instantaneous rate of change?
Average rate of change measures change over an interval, while instantaneous rate of change (derivative) measures change at a single point.
Can the average rate of change be undefined?
Yes, if the two x-values are the same (a = b), division by zero occurs and the rate is undefined.
Disclaimer: This calculator is for educational purposes. Verify results for critical applications.